Order Spherical Harmonic Expansion Research Articles

Recovery of layered tissue optical properties from spatial frequency-domain spectroscopy and a deterministic radiative transport solver.

.We present a method to recover absorption and reduced scattering spectra for each layer of a two-layer turbid media from spatial frequency-domain spectroscopy data. We focus on systems in which the thickness of the top layer is less than the transport mean free path . We utilize an analytic forward solver, based upon the ’th-order spherical harmonic expansion with Fourier decomposition method in conjunction with a multistage inverse solver. We test our method with data obtained using spatial frequency-domain spectroscopy with 32 evenly spaced wavelengths within to 1000 nm on six-layered tissue phantoms with distinct optical properties. We demonstrate that this approach can recover absorption and reduced scattering coefficient spectra for both layers with accuracy comparable with current Monte Carlo methods but with lower computational cost and potential flexibility to easily handle variations in parameters such as the scattering phase function or material refractive index. To our knowledge, this approach utilizes the most accurate deterministic forward solver used in such problems and can successfully recover properties from a two-layer media with superficial layer thicknesses.

Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements

We propose the P N approximation based on a finite element framework for solving the radiative transport equation with optical tomography as the primary application area. The key idea is to employ a variable order spherical harmonic expansion for angular discretization based on the proximity to the source and the local scattering coefficient. The proposed scheme is shown to be computationally efficient compared to employing homogeneously high orders of expansion everywhere in the domain. In addition the numerical method is shown to accurately describe the void regions encountered in the forward modeling of real-life specimens such as infant brains. The accuracy of the method is demonstrated over three model problems where the P N approximation is compared against Monte Carlo simulations and other state-of-the-art methods.

An adaptive finite element approach for neutron transport equation

In this paper, we develop an adaptive element refinement strategy that progressively refines the elements in appropriate regions of domain to solve even-parity Boltzmann transport equation. A posteriori error approach has been used for checking the approximation solutions for various sizes of elements. The local balance of neutrons in elements is utilized as an error assessment. To implement the adaptive approach a new neutron transport code FEMPT, finite element modeling of particle transport, for arbitrary geometry has been developed. This code is based on even-parity spherical harmonics and finite element method. A variational formulation is implemented for the even-parity neutron transport equation for the general case of anisotropic scattering and sources. High order spherical harmonic functions expansion for angle and finite element method in space is used as trial function. This code can be used to solve the multi-group neutron transport equation in highly complex X– Y geometries with arbitrary boundary condition. Due to powerful element generator tools of FEMPT, the description of desired and complicated 2D geometry becomes quite convenient. The numerical results show that the locally adaptive element refinement approach enhances the accuracy of solution in comparison with uniform meshing approach.

Magnetic fields and differential rotation on the pre-main sequence - I. The early-G star HD 141943 - brightness and magnetic topologies

Spectroscopic and spectropolarimetric observations of the pre-main sequence early-G star HD 141943 were obtained at four observing epochs (in 2006, 2007, 2009 and 2010). The observations were undertaken at the 3.9-m Anglo-Australian Telescope using the UCLES echelle spectrograph and the SEMPOL spectropolarimeter visitor instrument. Brightness and surface magnetic field topologies were reconstructed for the star using the technique of least-squares deconvolution to increase the signal-to-noise of the data. The reconstructed brightness maps show that HD 141943 had a weak polar spot and a significant amount of low latitude features, with little change in the latitude distribution of the spots over the 4 years of observations. The surface magnetic field was reconstructed at three of the epochs from a high order (l <= 30) spherical harmonic expansion of the spectropolarimetric observations. The reconstructed magnetic topologies show that in 2007 and 2010 the surface magnetic field was reasonably balanced between poloidal and toroidal components. However we find tentative evidence of a change in the poloidal/toroidal ratio in 2009 with the poloidal component becoming more dominant. At all epochs the radial magnetic field is predominantly non-axisymmetric while the azimuthal field is predominantly axisymmetric with a ring of positive azimuthal field around the pole similar to that seen on other active stars.

Lunar topographic model CLTM-s01 from Chang’E-1 laser altimeter

More than 3 million range measurements from the Chang’E-1 Laser Altimeter have been used to produce a global topographic model of the Moon with improved accuracy. Our topographic model, a 360th degree and order spherical harmonic expansion of the lunar radii, is designated as Chang’E-1 Lunar Topography Model s01 (CLTM-s01). This topographic field, referenced to a mean radius of 1738 km, has an absolute vertical accuracy of approximately 31 m and a spatial resolution of 0.25° (∼7.5 km). This new lunar topographic model has greatly improved previous models in spatial coverage, accuracy and spatial resolution, and also shows the polar regions with the altimeter results for the first time. From CLTM-s01, the mean, equatorial, and polar radii of the Moon are 1737103, 1737646, and 1735843 m, respectively. In the lunar-fixed coordinate system, this model shows a COM/COF offset to be (−1.777, −0.730, 0.237) km along the x, y, and z directions, respectively. All the basic lunar shape parameters derived from CLTM-s01 are in agreement with the results of Clementine GLTM2, but CLTM-s01 offers higher accuracy and reliability due to its better global samplings.

Toward High Throughput 3D Virtual Screening Using Spherical Harmonic Surface Representations

Searching chemical databases for possible drug leads is often one of the main activities conducted during the early stages of a drug development project. This article shows that spherical harmonic molecular shape representations provide a powerful way to search and cluster small-molecule databases rapidly and accurately. Our clustering results show that chemically meaningful clusters may be obtained using only low order spherical harmonic expansions. Our database search results show that using low order spherical harmonic shape-based correlation techniques could provide a practical and efficient way to search very large 3D molecular databases, hence leading to a useful new approach for high throughput 3D virtual screening. The approach described is currently being extended to allow the rapid search and comparison of arbitrary combinations of molecular surface properties.

Topography of the Moon from the Clementine lidar

Range measurements from the lidar instrument carried aboard the Clementine spacecraft have been used to produce an accurate global topographic model of the Moon. This paper discusses the function of the lidar; the acquisition, processing, and filtering of observations to produce a global topographic model; and the determination of parameters that define the fundamental shape of the Moon. Our topographic model; a 72nd degree and order spherical harmonic expansion of lunar radii, is designated Goddard Lunar Topography Model 2 (GLTM 2). This topographic field has an absolute vertical accuracy of approximately 100 m and a spatial resolution of 2.5°. The field shows that the Moon can be described as a sphere with maximum positive and negative deviations of ∼8 km, both occurring on the farside, in the areas of the Korolev and South Pole‐Aitken (S.P.‐Aitken) basins. The amplitude spectrum of the topography shows more power at longer wavelengths as compared to previous models, owing to more complete sampling of the surface, particularly the farside. A comparison of elevations derived from the Clementine lidar to control point elevations from the Apollo laser altimeters indicates that measured relative topographic heights generally agree to within ∼200 m over the maria. While the major axis of the lunar gravity field is aligned in the Earth‐Moon direction, the major axis of topography is displaced from this line by approximately 10° to the east and intersects the farside 24° north of the equator. The magnitude of impact basin topography is greater than the lunar flattening (∼2 km) and equatorial ellipticity (∼800 m), which imposes a significant challenge to interpreting the lunar figure. The floors of mare basins are shown to lie close to an equipotential surface, while the floors of unflooded large basins, except for S.P.‐Aitken, lie above this equipotential. The radii of basin floors are thus consistent with a hydrostatic mechanism for the absence of significant farside maria except for S.P.‐Aitken, whose depth and lack of mare require significant internal compositional and/or thermal heterogeneity. A macroscale surface roughness map shows that roughness at length scales of 101–102 km correlates with elevation and surface age.

Simulation of semiconductor devices using a Galerkin/spherical harmonic expansion approach to solving the coupled Poisson-Boltzmann system

This paper describes a Galerkin/spherical harmonics approach for solving the coupled Poisson-Boltzmann system of equations for the electron distribution function and the electric potential, which can then be used to calculate other parameters of interest such as current flow and electron temperature. The Galerkin approach described here has some pragmatic advantages in space-dependent problems over more commonly used term-matching techniques for arbitrary order spherical harmonic expansions in momentum space, but the method requires a careful treatment of the boundary conditions and upwinded discretization methods. Results are presented for nonuniformly doped one-dimensional devices using up to third order spherical harmonics to show the importance of including higher order harmonics to accurately calculate the distribution function in high field regions.

Discontinuous finite element solutions for neutron transport in X-Y geometry

A finite element method of composite solutions is described for the even-parity transport equation which uses a high order spherical harmonic expansions for the directional component of transport where they are needed and not everywhere as in established finite element methods. Previously in the method of composite solution trial functions were needed for both even and odd parity angular fluxes. For some problems there is a small saving in the number of unknowns used to specify a variational solution but for others savings of the order of 70% can be achived. The method is illustrated by numerical solutions in two dimensions for a Fletcher test problem, a PWR cell problem of Natelson and a problem of a quarter of a reactor with three control rods.

The figure of Phobos

A sixth degree and order spherical harmonic expansion was used to model the Phobos figure. This model appears to fit the entire surface to an accuracy of a few hundred meters, except in the region around the large crater Stickney. The model was validated using control points, limbs, terminators, map projections and predicted rotational properties. A for A forced libration amplitude of 0.81 ± 0.5 (1 σ) deg is predicted, matching the observed value of 0.79 deg which supports Phobos having a uniform density distribution to within the uncertainty of the prediction.

On utility of space-borne vector magnetic measurements in crustal studies

The MAGSAT data over India and contiguous regions for quiet periods ( K p ⩽ 3) have been analysed to isolate components of crustal origin. The main field contribution was removed with MGST ( 4 81 ) 13th degree and order spherical harmonic expansion. The external current contributions were estimated using the H- and Z-components directly and assuming the potential to have P 1 (cos θ) dependence. A quadratic polynomial for each satellite pass had to be removed to make residuals from adjacent passes agree with each other. These residuals over 2° × 2° blocks were averaged to prepare the anomaly maps. The anomaly maps exhibit better resolution than Langel et al. Features associated with the Indian Ocean geoid low clearly appear in the results. A test on internal consistency of reduction techniques was made by synthesizing B- map through the projection of three vector maps and comparing the same with the map obtained from direct measurements. It is noticed that near the dip equator the features of Z- map are not reflected in the B- map. We conclude that in those regions where the main field direction is such that one or two of its components X, Y or Z become zero or very small, the crustal anomaly in that direction will not influence the B- anomaly for it would have no projection in the direction of the main field.

The use of a non-spherical reference potential in statistical mechanical perturbation theory

The statistical perturbation theory for the pair correlation function using a non-spherical reference potential proposed earlier is studied further and evaluated here. First, we show that for even such a strongly quadrupolar fluid as liquid bromine, relatively few terms of the multiply-infinite set in the first order perturbation theory make a significant contribution. This greatly simplifies the calculations. Next, by comparing the results of the first order perturbation theory with our own Monte Carlo simulation studies for both reference and perturbed systems, we conclude that the first order perturbation theory using a non-spherical reference potential is rapidly convergent for the centre of mass (or orientation averaged) pair correlation function even for liquid bromine. However, the theory appears to converge less rapidly for the higher order spherical harmonic expansion coefficients of the angle dependent pair correlation function.

Radiative transfer via spherical harmonics

Abstract It is shown that several approximate treatments of radiative transfer (classic diffusion; diffusion with temperature jump boundary conditions; differential approximation) have a common origin in the spherical harmonic expansion of the equation of transfer. Comparisons with exact results, computed numerically with a discrete ordinates computer code (ANISN), are made for several problems, with and without heat sources, for the standard three one dimensional geometries. It is found that the lowest order spherical harmonic expansion (P-1, equivalent to the differential approximation) is superior to the next order expansion (P-2) for optically small systems. For intermediate and large systems, the P-2 expansion is more accurate. The P-2 expansion is identical to the temperature jump treatment of Deissler for problems with a spatially uniform heat source. For problems with a spatially dependent heat source, the P-2 expansion differs from the Deissler treatment and appears to be more accurate. These conclusions as to the accuracy of the P-l and P-2 expansions are somewhat different from the analogous situation in neutron transport theory, where the P-l approximation is regarded as universally more accurate than the P-2 approximation.

On recurrence relations for multipole coefficients

General recurrence relations between the coefficients in thenth and (n+1)th order spherical harmonic multipole expansions are derived. The particular application presented here is the derivation of the equations concerned with representing the geomagnetic field by magnetic multipoles. The equations up to the 3rd order multipole are given as an example of the method. The main advantage in using these recurrence relations rather than other methods is that the mathematics is reduced to merely a matter of successive substitutions and this allows a fast step by step generation of the required equations, in a form for which there is a simple numerical program for solution.

A new method of computing cosmic-ray cutoff rigidity for several geomagnetic field models

An approximate method of computing cosmic-ray cutoff rigidities is presented. The geomagnetic field is represented by the combination of a sixth order magnetic field whose origin lies entirely within the earth, and that due to a filamentary equatorial ring current of various radii and magnetic moments. The sixth order spherical harmonic expansion of the scalar magnetic potential is that obtained by Finch and Leaton by numerical analysis of surface magnetic field measurements for epoch 1955. Vertical cutoff rigidities have been computed for all IGY stations of geomagnetic latitude greater than ±45° listed in the Annals of the International Geophysical Year, vol. 8. In addition, the vertical cutoff rigidities have been calculated for geographic latitude steps of 5° in geographic longitude steps of 10° for the sixth order field alone in regions of geomagnetic latitude greater than ±45°. The results are compared with the values given by Quenby and Webber, and by Quenby and Wenk, and with those derived from the magnetic shell parameter L. Comparisons of the values of vertical cutoff rigidity obtained for Port aux Francais, Northern Italy, and Minneapolis with those obtained by McCracken, Lynch, and Kellogg, respectively, by calculation of particle trajectories, are consistent with the 5 per cent precision claimed for the present method.